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A Smooth, Recurrent, Non-Periodic Viscosity Solution of the Hamilton-Jacobi Equation

Skander Charfi

TL;DR

The paper tackles the lack of convergence in time‑dependent Hamilton–Jacobi dynamics by constructing a Mañé Lagrangian on any closed manifold of dimension at least two that yields a smooth, recurrent, non‑periodic viscosity solution to the Hamilton–Jacobi equation. The authors build periodic Peierls barriers with increasingly large periods and form an initial data $u$ as the infimum of these barriers, then refine this to a $C^ ablafty$‑regular solution $u_c$ via carefully tuned constants and symmetry constraints. They show that the resulting solution is recurrent but not periodic, with its omega‑limit set carrying adding‑machine (odometer) dynamics, and they describe the non‑wandering set of the Lax–Oleinik semigroup in this framework. The work integrates weak‑KAM/Aubry–Mather theory to characterize the omega‑limit sets and demonstrates that, under certain Mather–Mañé configurations, the Lax–Oleinik operator acts like an odometer on the non‑wandering set, highlighting rich long‑time behavior in non‑autonomous Hamilton–Jacobi dynamics. Overall, this provides a constructive approach to obtain smooth, non‑periodic recurrent viscosity solutions in higher dimensions and clarifies the asymptotic dynamics of the Lax–Oleinik semigroup in this regime.

Abstract

Viscosity solutions of the Hamilton-Jacobi equation were introduced by Lions and Crandall. For Tonelli Hamiltonians, these solutions are generated by the Lax-Oleinik operator. It is known that this operator converges in the autonomous framework, but this convergence fails in the general cases. In this paper, we introduce a method to construct smooth, recurrent, non-periodic viscosity solutions on fixed compact manifolds $M$ of dimension 2 or higher. Additionally, we provide a detailed description of the non-wandering set of the Lax-Oleinik operator and identify its action on various omega-limit sets.

A Smooth, Recurrent, Non-Periodic Viscosity Solution of the Hamilton-Jacobi Equation

TL;DR

The paper tackles the lack of convergence in time‑dependent Hamilton–Jacobi dynamics by constructing a Mañé Lagrangian on any closed manifold of dimension at least two that yields a smooth, recurrent, non‑periodic viscosity solution to the Hamilton–Jacobi equation. The authors build periodic Peierls barriers with increasingly large periods and form an initial data as the infimum of these barriers, then refine this to a ‑regular solution via carefully tuned constants and symmetry constraints. They show that the resulting solution is recurrent but not periodic, with its omega‑limit set carrying adding‑machine (odometer) dynamics, and they describe the non‑wandering set of the Lax–Oleinik semigroup in this framework. The work integrates weak‑KAM/Aubry–Mather theory to characterize the omega‑limit sets and demonstrates that, under certain Mather–Mañé configurations, the Lax–Oleinik operator acts like an odometer on the non‑wandering set, highlighting rich long‑time behavior in non‑autonomous Hamilton–Jacobi dynamics. Overall, this provides a constructive approach to obtain smooth, non‑periodic recurrent viscosity solutions in higher dimensions and clarifies the asymptotic dynamics of the Lax–Oleinik semigroup in this regime.

Abstract

Viscosity solutions of the Hamilton-Jacobi equation were introduced by Lions and Crandall. For Tonelli Hamiltonians, these solutions are generated by the Lax-Oleinik operator. It is known that this operator converges in the autonomous framework, but this convergence fails in the general cases. In this paper, we introduce a method to construct smooth, recurrent, non-periodic viscosity solutions on fixed compact manifolds of dimension 2 or higher. Additionally, we provide a detailed description of the non-wandering set of the Lax-Oleinik operator and identify its action on various omega-limit sets.
Paper Structure (45 sections, 72 theorems, 247 equations, 1 figure)

This paper contains 45 sections, 72 theorems, 247 equations, 1 figure.

Table of Contents

  1. Overview
  2. Main Result
  3. Structure of the Article
  4. Preliminary Tools and Properties
  5. Tonelli Hamiltonians and the Lax-Oleinik Operator
  6. Existence of Minimizing Curves
  7. Non-Expansiveness and Regularizing Properties of $\mathcal{T}$
  8. Calibrated Curves
  9. Peierls Barrier
  10. The Mather Set
  11. The Mañé Set
  12. Construction of the Mañé Hamiltonian
  13. Construction of the Lagrangian $L$
  14. Subdivision of the Manifold $M$
  15. The Isotopy $f_t$
  16. ...and 30 more sections

Key Result

Theorem 1

(Fathi MR1650261) For all scalar maps $u \in \mathcal{C}(M, \mathbb{R})$, the sequence $\mathcal{T}^nu$ converges in the $C^0$-topology to a weak-KAM solution $u_\infty$. Moreover, for all time $t >0$, $\mathcal{T}^t u_\infty = u_\infty$.

Figures (1)

  • Figure 1: The dynamics of the flow $g_t$ on the set $C_n$

Theorems & Definitions (159)

  • Theorem 1
  • Theorem 2
  • Theorem 1
  • Definition 1.1
  • Remark 1.2
  • Definition 1.3
  • Remark 1.4
  • Lemma 1.5
  • Definition 1.6
  • Proposition 1.7
  • ...and 149 more